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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 99
Chapter 6, Problem 99

Solve each problem. See Examples 5 and 9. The sum of two numbers is 47, and the difference between the numbers is 1. Find the numbers.

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Let the two numbers be represented as \(x\) and \(y\).
Write the system of equations based on the problem statement: \(x + y = 47\) and \(x - y = 1\).
Add the two equations to eliminate \(y\): \((x + y) + (x - y) = 47 + 1\) which simplifies to \(2x = 48\).
Solve for \(x\) by dividing both sides of the equation by 2: \(x = \frac{48}{2}\).
Substitute the value of \(x\) back into one of the original equations (for example, \(x + y = 47\)) to solve for \(y\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. In this problem, the sum and difference of two numbers form a system that can be solved together.
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Setting Up Equations from Word Problems

Translating a word problem into mathematical equations involves identifying variables and expressing relationships described in words as algebraic equations. Here, the sum and difference statements translate into two equations involving the two unknown numbers.
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Solving Systems by Addition or Substitution

Common methods to solve systems include addition (elimination) and substitution. Addition involves adding or subtracting equations to eliminate a variable, while substitution solves one equation for a variable and substitutes into the other. Both methods help find the values of the unknown numbers.
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Related Practice
Textbook Question

Solve each problem. See Examples 5 and 9. A cashier has a total of 30 bills, made up of ones, fives, and twenties. The number of twenties is 9 more than the number of ones. The total value of the money is \$351. How many of each denomination of bill are there?

Textbook Question

Find AB and BA for the following matrices.

A=[abcd]andB=[1001]A = \(\left\)[ \(\begin{matrix}\) a & b \\ c & d \(\end{matrix}\) \(\right\)] \(\quad\) \(\text{and}\) \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 1 & 0 \\ 0 & 1 \(\end{matrix}\) \(\right\)]

Matrix B acts as the multiplicative element for 2 ×\(\times\) 2 square matrices.

Textbook Question

Find the inverse, if it exists, for each matrix.

[2153]\(\left\)[ \(\begin{matrix}\) 2 & 1 \\ 5 & 3 \(\end{matrix}\) \(\right\)]

Textbook Question

Find the inverse, if it exists, for each matrix.

[2−101011−20]\(\left\)[ \(\begin{matrix}\) 2 & -1 & 0 \\ 1 & 0 & 1 \\ 1 & -2 & 0 \(\end{matrix}\) \(\right\)]

Textbook Question

Solve each problem. See Examples 5 and 9. The sum of the measures of the angles of any triangle is 180°. In a certain triangle, the largest angle measures 55° less than twice the medium angle, and the smallest angle measures 25° less than the medium angle. Find the measures of all three angles.

Textbook Question

Solve each problem. See Examples 5 and 9. At the Berger ranch, 6 goats and 5 sheep sell for \$305, while 2 goats and 9 sheep sell for \$285. Find the cost of a single goat and of a single sheep.